a Notice that the set that represents the animals which have gills will involve the set which represents the fish. Moreover, turtles will not be included in these sets because they do not have gills.
B
b Since the second statement and the conclusion are the negations of the hypothesis and the conclusion of the conditional statement, the contrapositive of the statement to explain why the argument uses good reasoning.
A
aVenn Diagram:
B
b See solution.
Practice makes perfect
a The following statements and the conclusion are given.
Given: &If an animal is a fish, then it has gills.
& A turtle does not have gills.
You conclude: &A turtle is not a fish.
If we define the Venn diagram of the given information, we can say that the set that represents the animals which have gills will involve the set which represents the fish. Moreover, turtles will not be included in these sets because they do not have gills.
b Let's represent the given information an the conclusion using p and q.
Statement
Representation
If an animal is a fish, then it has gills.
If p, then q.
A turtle does not have gills.
~ q
A turtle is not a fish.
~ p
Since the second statement and the conclusion are the negations of the hypothesis and the conclusion of the conditional statement, we will write the contrapositive of the statement to explain why the argument uses good reasoning.
Conditional: &If p, then q.
Contrapositive: &If ~ q, then ~ p.
We know that the truth value of a conditional and its contrapositive is the same. Therefore, we can make a valid conclusion using the Law of Detachment as the following.
If ~ q → ~ p &is true
and ~ q &is true,
then ~ p &is true.
As a result, the argument uses a good reasoning because it makes a conclusion by the Law of Detachment.
